which statement describes a key feature of function $g$ if $g(x) = f(x - 4)$?

horizontal asymptote of $y = -4$
horizontal asymptote of $y = 0$
$y$-intercept at $(0, 3)$
$y$-intercept at $(0, 1)$

Question

which statement describes a key feature of function $g$ if $g(x) = f(x - 4)$?

horizontal asymptote of $y = -4$
horizontal asymptote of $y = 0$
$y$-intercept at $(0, 3)$
$y$-intercept at $(0, 1)$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Horizontal Translation", "Exponential Functions", "Function Transformations" ], "new_concepts": [], "current_concepts": [ "Horizontal Translation", "Exponential Functions", "Function Transformations", "Asymptotes" ] } ## Identify the parent function and its features $$ \begin{aligned} &f(x) = 2^x\ &\text{Horizontal asymptote of } f(x): y = 0\ &\text{y-intercept of } f(x): (0, 1) \end{aligned} $$ ## Apply the horizontal translation $$ \begin{aligned} &g(x) = f(x - 4) = 2^{x-4}\ &\text{Transformation: Shift right by 4 units} \end{aligned} $$ ## Determine key features of the transformed function $$ \begin{aligned} &\text{Horizontal asymptote of } g(x): y = 0\ &\text{y-intercept of } g(x): (0, 2^{-4}) = (0, \frac{1}{16}) \end{aligned} $$

horizontal asymptote of $y = -4$

horizontal asymptote of $y = 0$

y-intercept at $(0, 3)$

y-intercept at $(0, 1)$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Function Transformations" ] }

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Question

Explanation:

Identify the parent function and its features

Apply the horizontal translation

Determine key features of the transformed function

Answer: